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Answer:

The general form of a sine - function is (y = A\sin(Bx - C)+D), for the function (y=-3\sin(\frac{1}{3}x)), we have (A = - 3), (B=\frac{1}{3}), (C = 0), (D = 0).

The amplitude of the function (y = A\sin(Bx)) is (|A|). Here, (|A|=| - 3| = 3), so the graph oscillates between (y=-3) and (y = 3).

The period of the function (y=\sin(Bx)) is given by (T=\frac{2\pi}{|B|}). For (y=-3\sin(\frac{1}{3}x)), since (B=\frac{1}{3}), the period (T=\frac{2\pi}{\frac{1}{3}}=6\pi).

1. Analyze the amplitude:
   • The amplitude of (y=-3\sin(\frac{1}{3}x)) is 3, which means the maximum value of the function is 3 and the minimum value is - 3. This eliminates some graphs that have incorrect maximum - minimum values.
2. Analyze the period:
   • The period (T = 6\pi). A standard sine function (y=\sin(x)) has a period of (2\pi). The function (y=\sin(\frac{1}{3}x)) has a period of (6\pi) (because when (B=\frac{1}{3}), (T=\frac{2\pi}{B}=6\pi)). This means the graph of (y=-3\sin(\frac{1}{3}x)) will complete one full cycle over an (x) - interval of length (6\pi).

The correct graph is the one that oscillates between (y = 3) and (y=-3) and has a period of (6\pi). Among the given options, the graph that has an amplitude of 3 and a period of (6\pi) is the first graph (the one where the graph oscillates between (y = 3) and (y=-3) and has a relatively wide - spaced wave pattern compared to the others).

Explanation:

Step1: Determine the amplitude

The amplitude (|A|) of (y=-3\sin(\frac{1}{3}x)) is (| - 3|=3).

Step2: Determine the period

Using the formula (T=\frac{2\pi}{|B|}), with (B = \frac{1}{3}), we get (T=\frac{2\pi}{\frac{1}{3}}=6\pi).

Step3: Match the graph

The graph with amplitude 3 and period (6\pi) is the correct one.